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  • DPRIETI Discussion Paper Series 10-E-002

    Knowledge Spillover on Complex Networks

    KONNO Tomohikothe University of Tokyo

    The Research Institute of Economy, Trade and Industry

  • 1

    RIETI Discussion Paper Series 10-E-002

    Knowledge Spillover on Complex Networks Tomohiko Konno

    Graduate School of Economics, The University of Tokyo Research Fellow of the Japan Society for the Promotion of Science


    Most growth theories have focused on R&D activities. Although R&D significantly influences

    economic growth, the spillover effect also has a considerable influence. In this paper, we study

    knowledge spillover among agents by representing it as network structures. The objective of this

    study is to construct a framework to treat knowledge spillover as a network. We introduce a

    knowledge spillover equation, solve it analytically to find a workable solution. It has mainly three

    properties: (1) the growth rate is common for all the agents only if they are linked to the entire

    network regardless of degrees, (2) the TFP level is proportional to degree, and (3) the growth rate is

    determined by the underlying network structure. We compare growth rate among representative

    networks: regular, random, and scale-free networks, and find the growth rate is the greatest in

    scale-free network. We apply this framework, i.e., knowledge spill over equation, to the problem of

    firms forming a network endogenously and show how distance and region size affect the economic

    growth. We also apply the framework to network formation mechanism. The aim of our paper is not

    just showing results, but in constructing a framework to study spillover by network.

    Keywords: Networks; Spillover; Economic Growth

    JEL Classification: O4; R11; R12

    *T.K. thanks to Takatoshi Tabuchi and Hiroshi Yoshikawa for helpful comments.

    RIETI Discussion Papers Series aims at widely disseminating research results in the form of professional papers, thereby stimulating lively discussion. The views expressed in the papers are solely those of the author(s), and do not present those of the Research Institute of Economy, Trade and Industry.

  • 1 Introduction

    Growth theories (Solow et al. (1957); Romer (1990); Aghion and Howitt(1998); Grossman and Helpman (1991)) show that the progress of technologydetermines the long-term growth. Most growth theories have focused onR&D activities. Although R&D significantly influences economic growth,the spillover effect also has a considerable influence. In this paper, we studyknowledge spillover among agents by representing it as network structuresand use complex network theories.

    Although it is occasionally assumed, mainly for simplicity, that once newtechnology is invented, it spreads worldwide immediately at no cost, tech-nology diffusion takes time and incurs various costs beyond any doubt. Thefollowing are just a few examples:

    It took a millennium for the water mill to be widely adoptedin Europe; it is felt that the main reason for this slow paceof diffusion was the absence of significant mobility duringpre-medieval and medieval times.

    The spread of new hybrid seed has been central to the in-crease in agricultural productivity over the past century.The classical work by Ryan and Gross (1943) documentsthat hybrid corn seed were adopted over a period of severalyears in the early twentieth century, in the United States.Moreover, diffusion of these seed displayed clear spatial pat-terns; initially, a small group of farmers adopted the seed,followed by their neighbors adopting it, and this was fol-lowed by the neighbors of the neighbors adopting it, and soon.

    The examples above are taken from Goyal (2007). The classic paper byGriliches (1957) shows that even more productive hybrid corn diffused onlyslowly in the U.S and the diffusion process was affected by the local eco-nomic conditions. He also found that technology diffusion can be describedby the logistic curve, occasionally referred to as the S-shaped curve. Ini-tially, it spreads only slowly, but once adoption reaches the critical point, itbegins to spread very rapidly; finally after a large fraction adopts, the rateof adoption declines. Griliches shows that the diffusion process takes time.Recent reviews by Asheim and Gertler (2005) also shows that geographicalproximity is an important factor for spillovers and Konno (2008) shows thatthe transaction between firms decreases as the distance increases.

    There is no doubt that technology difference exists across countries, fur-thermore the difference is not only across countries but also within a single


  • country. We can observe significant differences across firms in even a nar-rowly defined industry, Bartelsman et al. (2000) shows that technology islocal rather than global. Many studies find correlations between productiv-ity and firm size, various measures of technology (e.g, IT technology level),skill level of the employee, management practices, and so on. However, thequestion why there exists a significant productivity difference among firmswithin a single country and within even a narrowly defined industry is not yetanswered. Therefore, it is not surprising that we still lack a consensus on whatdetermines cross-country productivity differences. These evidences show thatnew technology does not spread instantaneously and technology difference ex-ists, and in particular it suggests that knowledge spillover structure surelyexist, which we will express by network structures. Unless such networks ex-isted meaning that technology diffuses world wide instantaneously, it wouldbe difficult to explain why such a significant technology difference exists.

    The following recent papers study knowledge spillover. Keller (2004) dis-cussing spillovers and geographical relation; Eaton and Kortum (2001) show-ing convergence, spillovers and trade; Eaton and Kortum (1999); Acemogluet al. (2006); and Vandenbussche et al. (2006). Knowledge spillover mustbe related to geography; in this respect Fujita and Thisse (2002), SpatialEconomics, may also be relevant.

    Our study also focuses on the network structure, placing it among otherstudies on networks. Goyal and Moraga-Gonzalez (2001) studied the R&Dformation mechanism by which coalition diminishes marginal cost, while ourmodel directly deals with knowledge spillover which increases TFP and fo-cuses on the processes of technology diffusion by network structure of thatprocess. For the Economics of networks, please refer Goyal (2007); Jackson(2008).

    Before introducing our model, we briefly explain the standard model thatanalyzes knowledge spillover. For convenience, we explain it with the modelof the international world technology frontier by following Acemoglu (2009).The world consists of J countries indexed by j = 1, 2, , J . Each countryhas the following production function:

    Yj(t) = F (Kj(t), Aj(t)Lj(t)) (1)

    We define growth rate of the country j, gj, by

    gj(t) Aj(t)


    Let us assume that world technology frontier, which is denoted by A(t), growsexogenously at the constant rate


  • g A(t)A(t)


    The population growth is ignored, then the utility function is

    Uj =



    1 11

    ]dt (4)

    where cj Cj(t)/Lj(t) is the per capita consumption in country j at time t.We assume that is the same across all the countries1.

    As in the neoclassical growth model, the flow of capital is described by

    kj(t) = f(kj(t)) cj(t) ( + gj(t))kj(t) (5)

    where cj(t) cj(t) Cj(t)/Aj(t)Lj(t) is the consumption normalized byeffective units of labor.

    In this model, knowledge spillover is described by the following equation:

    Aj(t) = j (A(t) Aj(t)) + jAj(t) (6)

    Eq.(6) states that each country absorbs world technology at the exogenousconstant rate j. If the country j is far behind the world technology frontier,then Aj(t) grows faster. In contrast, if Aj(t) = A(t), the country j hasnothing to learn from the world technology frontier.

    We also define aj(t) as

    aj(t) Aj(t)


    Then we can re-write Eq.(6) as

    aj(t) = j (j + g j) aj(t) (8)

    gj(t) becomes

    gj(t) =aj(t)

    aj+ g (9)

    There exists a unique steady state such that aj(t) = 0 j.

    aj =j

    j + g j(10)

    f (kj ) = + + g (11)

    1Do not confuse this with the which will be introduced into our model later andmeans self evolution rate.


  • and consumption per capita in each country grows at the constant rate g,which is the growth rate of world technology.

    In this type of models, the network structure of knowledge spillover isnot explicitly considered. There exist only two kinds of countries an ordi-nary country and one that is not actually a country; but a world technologyfrontier. Interactions among agents where knowledge diffuses from one agentto other agents are not explicitly considered. Instead of introducing such in-teractions among many countries, ordinary countries are affected only by theworld technology frontier. However, in this model, these countries do not af-fect other countries nor they are affected by them. This is probably because itis difficult to find an analytical solution where there is an asymmetric networkstructure of knowledge spillover in which degree distribution is heterogenous.It is also probably because this type of model was invented before ComplexNetworks emerged around 2000, when people realized that explicitly consid-ering the network structure in the model has significant importance. Studieson Complex Networks small-world and scale-free networks a


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